Appendix B Introduction to Graphing Calculators B1

Appendix B Introduction to Graphing Calculators

IntroductionYou previously studied the point-plotting method for sketching the graph of an equation. One of the disadvantages of the point-plotting method is that to get a good idea about the shape of a graph, you need to plot many points. By plotting only a few points, you can badly misrepresent the graph.

For instance, consider the equation y = x3. To graph this equation, suppose you calculated only the following three points.

x −1 0 1

y x3 −1 0 1

Solution point (−1, −1) (0, 0) (1, 1)

By plotting these three points, as shown in Figure B.1, you might assume that the graph of the equation is a line. This, however, is not correct. By plotting several more points, as shown in Figure B.2, you can see that the actual graph is not straight at all.

21−1

−1

−2

−2

2

1

(0, 0)

(1, 1)

x

y

(−1, −1)

21−1−2

1

2

x

y

−1

−2

y = x3

Figure B.1 Figure B.2

So, the point-plotting method leaves you with a dilemma. On the one hand, the method can be very inaccurate if only a few points are plotted. But, on the other hand, it is very time-consuming to plot a dozen (or more) points. Technology can help you solve this dilemma. Plotting several points (or even hundreds of points) on a rectangular coordinate system is something that a computer or graphing calculator can do easily.

Introduction

Using a Graphing Calculator

Using Special Features of a Graphing Calculator

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B2 Appendix B Introduction to Graphing Calculators

Use a graphing calculator to graph 2y + x = 4.

SolutIon

To begin, solve the equation for y in terms of x.

2y + x = 4 Write original equation.

2y = −x + 4 Subtract x from each side.

y = − 1 — 2 x + 2 Divide each side by 2.

Press the Y= key, and enter the following keystrokes.

(−) X, T, θ, n ÷ 2 + 2

The top row of the display should now be as follows.

Y1 = −X/2 + 2

Press the GRAPH key, and the screen should look like the one in Figure B.3.

Graphing a Linear Equation

using a Graphing CalculatorThere are many different graphing utilities: some are graphing programs for computers and some are hand-held graphing calculators. In this appendix, the steps used to graph an equation with a TI-84 Plus graphing calculator are described. Keystroke sequences are often given for illustration; however, these sequences may not agree precisely with the steps required by your calculator.*

*The graphing calculator keystrokes given in this appendix correspond to the TI-84 Plus graphing calculators by Texas Instruments. For other graphing calculators, the keystrokes may differ. Consult your user’s guide.

−10

−10

10

10

Figure B.3

Graphing an Equation with a TI-84 Plus Graphing Calculator

Before performing the following steps, set your calculator so that all of the standard defaults are active. For instance, all of the options at the left of the MODE screen should be highlighted.

1. Set the viewing window for the graph. (See Example 3.) To set the standard viewing window, press ZOOM 6.

2. Rewrite the equation so that y is isolated on the left side of the equation. 3. Press the Y= key. Then enter the right side of the equation on the first line

of the display. (The first line is labeled Y1 = .) 4. Press the GRAPH key.

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Appendix B Introduction to Graphing Calculators B3

Graphing an Equation Involving Absolute Value

Use a graphing calculator to graph

y = x − 3 .SolutIon

This equation is already written so that y is isolated on the left side of the equation. Pressthe Y= key, and enter the following keystrokes.

MATH ▶ 1 X, T, θ, n − 3 )

The top row of the display should now be as follows.

Y1 = abs(X − 3)

Press the GRAPH key, and the screen should look like the one shown in Figure B.5.

In Figure B.3, notice that the calculator screen does not label the tick marks on the x-axis or the y-axis. To see what the tick marks represent, you can press WINDOW . If you set your calculator to the standard graphing defaults before working Example 1, the screen should show the following values.

Xmin = −10 The minimum x-value is −10.

Xmax = 10 The maximum x-value is 10.

Xscl = 1 The x-scale is 1 unit per tick mark.

Ymin = −10 The minimum y-value is −10.

Ymax = 10 The maximum y-value is 10.

Yscl = 1 The y-scale is 1 unit per tick mark.

Xres = 1 Sets the pixel resolution

These settings are summarized visually in Figure B.4.

Ymin

XmaxXminXscl

Yscl

Ymax

Figure B.4

−10

−10

10

10

Figure B.5

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B4 Appendix B Introduction to Graphing Calculators

Setting the Viewing Window

Use a graphing calculator to graph

y = x2 + 12.

SolutIon

Press Y= and enter x2 + 12 on the first line.

X, T, θ, n x2 + 12

Press the GRAPH key. If your calculator is set to the standard viewing window, nothing will appear on the screen. The reason for this is that the lowest point on the graph of y = x2 + 12 occurs at the point (0, 12). Using the standard viewing window, you obtain a screen whose largest y-value is 10. In other words, none of the graph is visible on a screen whose y-values vary between −10 and 10, as shown in Figure B.6. To changethese settings, press WINDOW and enter the following values.

Xmin = −10 The minimum x-value is −10.

Xmax = 10 The maximum x-value is 10.

Xscl = 1 The x-scale is 1 unit per tick mark.

Ymin = −10 The minimum y-value is −10.

Ymax = 30 The maximum y-value is 30.

Yscl = 5 The y-scale is 5 units per tick mark.

Xres = 1 Sets the pixel resolution

Press GRAPH and you will obtain the graph shown in Figure B.7. On this graph, note that each tick mark on the y-axis represents five units because you changed the y-scale to 5. Also note that the highest point on the y-axis is now 30 because you changed the maximum value of y to 30.

−10

−10

10

10

−10

−10

10

30

Figure B.6 Figure B.7

using Special Features of a Graphing Calculator

To use your graphing calculator to its best advantage, you must learn to set the viewing window, as illustrated in the next example.

If you changed the y-maximum and y-scale on your calculator as indicated in Example 3, you should return to the standard setting before working Example 4. To do this, press ZOOM 6.

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Appendix B Introduction to Graphing Calculators B5

Using a Square Setting

Use a graphing calculator to graph y = x. The graph of this equation is a line that makes a 45° angle with the x-axis and with the y-axis. From the graph on your calculator, does the angle appear to be 45°?

SolutIon

Press Y= and enter x on the first line.

Y1 = X

Press the GRAPH key and you will obtain the graph shown in Figure B.8. Notice that the angle the line makes with the x-axis doesn’t appear to be 45°. The reason for this is that the screen is wider than it is tall. This makes the tick marks on the x-axis farther apart than the tick marks on the y-axis. To obtain the same distance between tick marks on both axes, you can change the graphing settings from “standard” to “square.” To do this, press the following keys.

ZOOM 5 Square setting

The screen should look like the one shown in Figure B.9. Note in this figure that the square setting has changed the viewing window so that the x-values vary from −15 to 15.

There are many possible square settings on a graphing calculator. To create a squaresetting, you need the following ratio to be 2 — 3 .

Ymax − Ymin —— Xmax − Xmin

For instance, the setting in Example 4 is square because

Ymax − Ymin —— Xmax − Xmin = 10 − (−10) — 15 − (−15) = 20 — 30 = 2 — 3 .

Graphing More than One Equation

Use a graphing calculator to graph each equation in the same viewing window.

y = −x + 4, y = −x, and y = −x − 4

SolutIon

To begin, press Y= and enter all three equations on the first three lines. The display should now be as follows.

Y1 = −X + 4 (−) X, T, θ, n + 4

Y2 = −X (−) X, T, θ, n

Y3 = −X − 4 (−) X, T, θ, n − 4

Press the GRAPH key and you will obtain the graph shown in Figure B.10. Note that the graph of each equation is a line and that the lines are parallel to each other.

−10

−10

10

10

Figure B.8

−15

−10

15

10

Figure B.9

−10

−10

10

10

Figure B.10

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B6 Appendix B Introduction to Graphing Calculators

Using the Trace Feature

Approximate the x- and y-intercepts of y = 3x + 6 by using the trace feature of a graphing calculator.

SolutIon

Press Y= and enter 3x + 6 on the first line.

3 X, T, θ, n + 6

Press the GRAPH key and you will obtain the graph shown in Figure B.11. Then pressthe TRACE key and use the ◀ ▶ keys to move along the graph. To get a better approximation of a solution point, you can use the following keystrokes repeatedly.

ZOOM 2 ENTER

As you can see in Figures B.12 and B.13, the x-intercept is (−2, 0) and the y-intercept is (0, 6).

−2.04

−0.044

−1.96

0.034

−10

−10

10

10

Figure B.12 Figure B.13

−10

−10

10

10

Figure B.11

Appendix B ExercisesIn Exercises 1–12, use a graphing calculator to graph the equation. (Use a standard setting.) See Examples 1 and 2.

1. y = −3x 2. y = x − 4

3. y = 3 — 4 x − 6 4. y = −3x + 2

5. y = 1 — 2 x2 6. y = − 2 — 3 x2

7. y = x2 − 4x + 2 8. y =−0.5x2 − 2x + 2

9. y = x − 5 10. y = x + 4 11. y = x2 − 4 12. y = x − 2 − 5

In Exercises 13–16, use a graphing calculator to graph the equation using the given window settings. See Example 3.

13. y = 27x + 100 14. y = 50,000 − 6000x

Xmin = 0 Xmin = 0 Xmax = 5 Xmax = 7 Xscl = .5 Xscl = .5 Ymin = 75 Ymin = 0 Ymax = 250 Ymax = 50000 Yscl = 25 Ysztcl = 5000 Xres = 1 Xres = 1

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Appendix B Introduction to Graphing Calculators B7

15. y = 0.001x2 + 0.5x 16. y = 100 − 0.5 x Xmin = −500 Xmin = −300 Xmax = 200 Xmax = 300 Xscl = 50 Xscl = 60 Ymin = −100 Ymin = −100 Ymax = 100 Ymax = 100 Yscl = 20 Yscl = 20 Xres = 1 Xres = 1

In Exercises 17–20, find a viewing window that shows the important characteristics of the graph.

17. y = 15 + x − 12 18. y = 15 + (x − 12)2

19. y = −15 + x + 12 20. y = −15 + (x + 12)2

Geometry In Exercises 21–24, use a graphing calculator to graph the equations in the same viewing window. Using a “square setting,” determine the geometrical shape bounded by the graphs. See Example 4.

21. y = −4, y = − x 22. y = x , y = 5

23. y = x − 8, y = − x + 8

24. y = − 1 — 2 x + 7, y = 8 — 3 (x + 5), y = 2 — 7 (3x − 4)

In Exercises 25–28, use a graphing calculator to graph both equations in the same viewing window. Are the graphs identical? If so, what basic rule of algebra is being illustrated? See Example 5.

25. y1 = 2x + (x + 1)

y2 = (2x + x) + 1

26. y1 = 1 — 2 (3 − 2x)

y2 = 3 — 2 − x

27. y1 = 2 ( 1 — 2 ) y2 = 1

28. y1 = x(0.5x)

y2 = (0.5x)x

In Exercises 29–36, use the trace feature of a graphing calculator to approximate the x- and y-intercepts of the graph. See Example 6.

29. y = 9 − x2

30. y = 3x2 − 2x − 5

31. y = 6 − x + 2 32. y = (x − 2)2 − 3

33. y = 2x − 5

34. y = 4 − x 35. y = x2 + 1.5x − 1

36. y = x3 − 4x

Modeling Data In Exercises 37 and 38, use the following models, which give the number of pieces of first-class mail and the number of pieces of standard mail handled by the U.S. Postal Service.

First Classy = 0.73x2 − 19.9x + 204, 8 ≤ x ≤ 11

Standardy = −2.437x3 + 74.16x2 − 748.5x + 2589, 8 ≤ x ≤ 11

In these models, y is the number of pieces handled (in billions) and x is the year, with x 8 corresponding to 2008. (Source: U.S. Postal Service)

37. Use the following window settings to graph both models in the same viewing window on a graphing calculator.

Xmin = 8 Xmax = 11 Xscl = 1 Ymin = 60 Ymax = 110 Yscl = 10 Xres = 1

38. (a) Were the numbers of pieces of first-class mail and standard mail increasing or decreasing over time?

(b) In what year were the numbers of pieces of first-class mail and standard mail the same?

(c) After the year in part (b), was more first-class mail or more standard mail handled?

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